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S18b. What is a vector?
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A vector is (for the beginner) a list of numbers written below each 
other. For example the x,y, and z coordinate of a point in a 
3-dimensional coordinate system. Physicists write the three 
coordinates as x_1, x_2, x_3 and combine it to a vector
simply called x.
          /   \
         | x_1 |
     x = | x_2 |  (The parentheses look a bit awkward in ascii.)
         | x_3 |
          \   /
The same for a list of n numbers. This gives a vector x with n 
coordinates x_1,...x_n, and is thought of as a point in a
space with n dimensions.

Two vectors are added or subtracted just by adding or subtracting 
their entries. A vector is multiplied by a number just by multiplying 
each entry with the number. Then there is the inner product of two 
vectors
     x dot y = sum_i x_i*y_i
which is a number and not a vector.

Once you mastered vectors you need to understand matrices.
These are rectangular arrays of numbers.

Later you need to enrich the meaning of a vector by learning
the concept of a vector space. Now all sorts of objects might
also deserve the name vector, most prominently functions, 
matrices, tensors, operators. They behave in many respects
just like ordinary vectors.